Measuring Solid Volume
Suppose, in the preceding problem, that the object in question is irregular. How can we know that its volume is 45.3 cm ^{3} ? It would be easy to figure out the volume if the object were a perfect sphere or a perfect cube or a rectangular prism. Suppose, however, that it’s a knobby little thing?
Scientists have come up with a clever way of measuring the volumes of irregular solids: Immerse them in a liquid. First, measure the amount of liquid in a container (Fig. 10-2a). Then measure the amount of liquid that is displaced when the object is completely submerged. This will show up as an increase in the apparent amount of liquid in the container (see Fig. 10-2b). One milliliter (1 ml) of water happens to be exactly equal to 1 cm ^{3} , and any good chemist is bound to have a few containers marked off in milliliters. This is the way to do it, then, provided the solid does not dissolve in the liquid and that none of the liquid is absorbed into the solid.
Fig. 10-2 . Measuring the volume of a solid, ( a ) a container with liquid but without the sample; ( b ) a container with the sample totally submerged in the liquid.
Specific Gravity Of Solids
Another important characteristic of a solid is its density relative to that of pure liquid water at 4°C (about 39°F). Water is at its most dense at this temperature and is assigned a relative density of 1. Substances with relative density greater than 1 will sink in pure water at 4°C, and substances with relative density less than 1 will float in pure water at 4°C. The relative density of a solid, defined in this way, is called the specific gravity . You often will see this abbreviated as sp gr. It is also known as relative density.
You certainly can think of substances whose specific gravity numbers are greater than 1. Examples include most rocks and virtually all metals. However, pumice, a volcanic rock that is filled with air pockets, floats on water. Most of the planets, their moons, and the asteroids and meteorites in our solar system have specific gravities greater than 1, with the exception of Saturn, which would float if a lake big enough could be found in which to test it!
Interestingly, water ice has specific gravity of less than 1, so it floats on liquid water. This property of ice is more significant than you might at first suppose. It allows fish to live underneath the frozen surfaces of lakes in the winter in the temperate and polar regions of the Earth because the layer of ice acts as an insulator against the cold atmosphere. If ice had specific gravity of greater than 1, it would sink to the bottoms of lakes during the winter months. This would leave the surfaces constantly exposed to temperatures below freezing, causing more and more of the water to freeze, until shallow lakes would become ice from the surface all the way to the bottom. In such an environment, all the fish would die during the winter because they wouldn’t be able to extract the oxygen they need from the solid ice, nor would they be able to swim around in order to feed themselves. It is difficult to say how life on Earth would have evolved if water ice had a specific gravity of greater than 1.
Elasticity Of Solids
Some solids can be stretched or compressed more easily than others. A piece of copper wire, for example, can be stretched, although a similar length of rubber band can be stretched much more. However, there is a difference in the stretchiness of these two substances that goes beyond mere extent. If you let go of a rubber band after stretching it, it will spring back to its original length, but if you let go of a copper wire, it will stay stretched.
The elasticity of a substance is the extent of its ability to return to its original dimensions after a sample of it has been stretched or compressed. According to this definition, rubber has high elasticity, and copper has low elasticity. Note that elasticity, defined in this way, is qualitative (it says something about how a substance behaves) but is not truly quantitative (we aren’t assigning specific numbers to it). Scientists can and sometimes do define elasticity according to a numerical scheme, but we won’t worry about that here. It is worth mentioning that there is no such thing as a perfectly elastic or perfectly inelastic material in the real world. Both these extremes are theoretical ideals.
This being said, suppose that there does exist a perfectly elastic substance. Such a material will obey a law concerning the extent to which it can be stretched or compressed when an external force is applied. This is called Hooke’s law: The extent of stretching or compression of a sample of any substance is directly proportional to the applied force. Mathematically, if F is the magnitude of the applied force in newtons and s is the amount of stretching or compression in meters, then
s = kF
where k is a constant that depends on the substance. This can be written in vector form as
s = k F
to indicate that the stretching or compression takes place in the same direction as the applied force.
Perfectly elastic stuff can’t be found in the real world, but there are plenty of materials that come close enough so that Hooke’s law can be considered valid in a practical sense, provided that the applied force is not so great that a test sample of the material breaks or is crushed.
Elasticity Of Solids Practice Problems
Problem
Suppose that an elastic bungee cord has near perfect elasticity as long as the applied stretching force does not exceed 5.00 N. When no force is applied to the cord, it is 1.00 m long. When the applied force is 5.00 N, the band stretches to a length of 2.00 m. How long will the cord be if a stretching force of 2.00 N is applied?
Solution
Applying 5.00 N of force causes the cord to become 1.00 m longer than its length when there is no force. We are assured that the cord is “perfectly elastic” as long as the force does not exceed 5.00 N. Therefore, we can calculate the value of the constant k , called the spring constant , in meters per newton (m/N) by rearranging the preceding formula:
s = kF
k = s/F
k = (1.00 m)/(5.00 N) = 0.200 m/N
provided that F ≤ 5.00 N. Therefore, the formula for displacement as a function of force becomes
s = 0.200 F
If F = 2.00 N, then
s = 0.200 m/N × 2.00 N = 0.400 m
This is the additional length by which the cord will “grow” when the force of 2.00 N is applied. Because the original length, with no applied force, is 1.00 m, the length with the force applied is 1.00 m + 0.400 m = 1.400 m. Theoretically, we ought to round this off to 1.40 m.
The behavior of this bungee cord, for stretching forces between 0 and 5.00 N, can be illustrated graphically as shown in Fig. 10-3 . This is a linear function; it appears as a straight line when graphed in standard rectangular coordinates. If the force exceeds 5.00 N, according to the specifications for this particular bungee cord, we have no assurance that the function of displacement versus force will remain linear. In the extreme, if the magnitude of the stretching force F is great enough, the cord will snap, and the displacement s will skyrocket to indeterminate values.
Fig. 10-3 . Illustration for Problem 10-3. The function is linear within the range of forces shown here.
Practice problems of these concepts can be found at: Basic States Of Matter Practice Test
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From Physics Demystified: A Self-Teaching Guide. Copyright © 2002 by The McGraw-Hill Companies, Inc. All Rights Reserved.